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After examining the "big picture, " it became evident that over the course of this 16-week period, significant transitional moments existed, during which collaborations among the dyads and the group seemed to change, and the quality of discourse improved for both groups. While not directly related to any specific problem type or context, these transitional moments seemed to be related to on-going negotiations and relationships, my role as a teacher/facilitator in the development of the dyads' effective listening (Davis, 1997) and to their beliefs and mathematical understandings. Prolonged problem solving and on-going negotiations and collaborations seemed to be related to students' experiences with productive interactions, shared authorities, and meaningful discourse as well as developing a supportive environment that was beneficial to its participants.

The purpose of this case study was to explore the complex interplay among student beliefs, problem solving engagement, problem type, and mathematics understanding as well as dynamics within group discourse among four ninth-grade mathematics students. The study aimed to provide insight into the relationship that exist between student engagement and problem type they choose to solve and to understand how their problem solving discourse evolve as students participate in a collaborative problem solving environment. This case study, focusing on two dyads' problem posing, problem solving, and collaboration, sheds light on the envisioning of curriculum alternatives for mathematics education amidst the many constraints of current and traditional problem solving contexts. The analysis of both these dyad's 16-week long collaboration reveals that the role of conversation, prolonged problem solving interactions, and on-going negotiations and relationships is key in their transformation.

The results suggest, as students developed a culture within their dyads, of problem solving and problem posing, and collaboration, that engagement was increased. After evaluating the various data relating to problem type and participant engagement, it became evident that certain problem types engaged the students more than the others. While it was no surprise that routine problems were not engaging to them, it was also evident that their collaboration and discourse were very different under these circumstances. They were much less likely to question, challenge, argue, negotiate, or probe each others' thinking, and were much more likely to rely on and accept the first answer. There were no efforts to modify, extend, or apply these routine problems to other contexts.

While this study did illuminate some limitations that a prolonged problem solving atmosphere might present in a traditional classroom setting, for the most part, the results of this work supported recent reform-based literature that advocates the use of nonroutine problems in a collaborative environment. This conclusion is equally supported by the enhanced level of mathematics conceptual understanding manifested in the participants at the close of this study, as well as changes that occurred in their pedagogical, collaborative, technical, and mathematics beliefs and assumptions.